Mean ke teen methods, Median aur Mode — complete derivations aur worked examples.
📋 Grouped Data Kya Hai?
🏫 School Exam Example:
Ek school mein 200 students ke marks 0 se 100 tak hain. Agar hum har student ka mark alag se list karein — 200 numbers — bahut mushkil ho jaayega patterns dekhna. Isliye hum class intervals banate hain: 0-10, 10-20, 20-30... aur count karte hain kitne students har range mein hain. Yahi grouped frequency distribution hai!
Important Terms: Class Interval: Range jaise 10-20 (lower = 10, upper = 20) Class Width (h): Upper − Lower = 20−10 = 10 Class Mark (xᵢ): Middle value = (10+20)/2 = 15 Frequency (fᵢ): Us class mein kitne data points hain Cumulative Frequency (cf): Current tak ki sari frequencies ka sum
Sample Data: Students ke Marks (50 students)
Class Interval
Frequency (fᵢ)
Class Mark (xᵢ)
fᵢxᵢ
Cumulative f
10-25
2
17.5
35
2
25-40
3
32.5
97.5
5
40-55
7
47.5
332.5
12
55-70
6
62.5
375
18
70-85
6
77.5
465
24
85-100
6
92.5
555
30
Total
Σf=30
Σfx=1860
📊 Mean — 3 Methods
Method 1: Direct Method
Mean = Σfᵢxᵢ / Σfᵢ
✅ Best when numbers are small and easy to calculate
Direct Method on above data:
Mean = 1860 / 30 = 62
Method 2: Assumed Mean Method (Short Cut)
Mean = a + (Σfᵢdᵢ / Σfᵢ)
dᵢ = xᵢ − a (a = assumed mean, usually middle class mark)
✅ Best when xᵢ values are large but differences dᵢ are small
Assumed Mean Method: a = 62.5 (middle class mark)
Class
fᵢ
xᵢ
dᵢ=xᵢ−62.5
fᵢdᵢ
10-25
2
17.5
−45
−90
25-40
3
32.5
−30
−90
40-55
7
47.5
−15
−105
55-70
6
62.5
0
0
70-85
6
77.5
+15
+90
85-100
6
92.5
+30
+180
Total
30
Σfᵢdᵢ = −15
Mean = 62.5 + (−15/30) = 62.5 − 0.5 = 62 ✓
Method 3: Step Deviation Method
Mean = a + h × (Σfᵢuᵢ / Σfᵢ)
uᵢ = (xᵢ − a) / h (divide by class width h)
✅ Best when class width h is constant — simplest calculations
Median: Data ko 2 equal halves mein divide karne wala value.
Grouped data mein hum formula use karte hain kyunki exact values pata nahi hain.
Median Formula
Median = l + [ (n/2 − cf) / f ] × h
l = lower boundary of median class
n = Σf = total frequency
cf = cumulative frequency BEFORE median class
f = frequency OF median class
h = class width
Median class kaise dhundein?
1. Cumulative frequency table banao
2. n/2 calculate karo (n = total frequency)
3. Jis class mein cumulative frequency n/2 ko cross kare — woh median class
Example: Upar wali table se Median nikalo (n=30, n/2=15)
l = lower boundary of modal class (class with highest frequency)
f₁ = frequency of modal class
f₀ = frequency of class BEFORE modal class
f₂ = frequency of class AFTER modal class
h = class width
Mode = 3 × Median − 2 × Mean
Verify: Mode = 3(62.5) − 2(62) = 187.5 − 124 = 63.5 ≈ 52 (ye approximately correct hoga, exact nahi kyunki yeh empirical formula hai)
📊 Ogive (Cumulative Frequency Graph)
Ogive = cumulative frequency ka graph. Do types: Less than Ogive: Upper class boundary vs cumulative frequency (badh ta jaata hai) More than Ogive: Lower class boundary vs (total−cumulative) frequency (ghat ta jaata hai)
Dono Ogive ka intersection = Median!
📝 Practice Questions
Q1. Neeche diya hua data hai. Direct method se mean nikalo.
Wages (₹)
Workers
100-120
10
120-140
15
140-160
20
160-180
22
180-200
18
200-220
15
xᵢ: 110,130,150,170,190,210. fᵢxᵢ: 1100,1950,3000,3740,3420,3150. Σfᵢxᵢ=16360. Σf=100. Mean = 163.6
Q2. Upar wali table se modal class kaunsi hai? Mode nikalo.